Electromechanical Dynamics (Part 1).0094

Electromechanical Dynamics (Part 1).0094 - shows the change...

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Electromechanical Coupling Then the energy equation (3.1.9) is dW,' = Adi + f" dx, (3.1.32) where W, = Ai - W,,. (3.1.33) The energy equation (3.1.32) now has the required form in which changes in the function W, are accounted for by changes in the independent variables (i, x). The function W'(i, x) is called the coenergy and is defined in terms of the energy Wj,(i, x) and terminal relations A(i, x) by (3.1.33).* Remember that (3.1.32) physically represents conservation of energy for the coupling network. The form of this equation is similar to that of (3.1.9) and our arguments now parallel those of Section 3.1.2a. Because W' = WA.(i, x), dW' = W di + W" dx. (3.1.34) di az We subtract (3.1.34) from (3.1.32) to obtain 0= - ) d f - dx. (3.1.35) ai dx' ax Because di and dx are independent (arbitrary), A= ', (3.1.36) f = aW"(i, x) (3.1.37) ax If the stored energy (hence coenergy) is known, the electrical and mechanical terminal relations can be calculated. Comparison of (3.1.37) and (3.1.23)
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Unformatted text preview: shows the change in the form of the force expression when the electrical variable chosen as independent is changed from 2 to i. The result of (3.1.37) can be generalized to a system with any number of terminal pairs in a straightforward manner (see Table 3.1). For a magnetic field system with N electrical terminal pairs and M translational mechanical terminal pairs the conservation of energy equation becomes NM dW, = , i, dij -, fj" dzj. (3.1.38) 1=1 j=1 We now use the generalization of (3.1.31), N N N Zij dA, = Xd(ijA) -A2, di 1 , (3.1.39) =1 j=i1 I t * This manipulation, which represents conservation of energy in terms of new independent variables, is called a Legendre transformation in classical mechanics and thermodynamics. 3.1.2 A-PDF Split DEMO : Purchase from www.A-PDF.com to remove the watermark...
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This note was uploaded on 02/10/2012 for the course MECE 4371 taught by Professor Liu during the Fall '11 term at University of Houston.

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